In the series: Note 9.

Version: 0.2

By: Albert van der Sel

Doc. Number: Note 9.

For who: for beginners.

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Status: Ready.

Maybe you need to pick up "some" basic "mathematics" rather

So really..., my emphasis is on "rather

So, I am really not sure of it, but I hope that this note can be of use.

Ofcourse, I hope you like my "style" and try the note anyway.

This note: Note 9: Complex numbers.

Each note in this series, is build "on top" of the preceding ones.

Please be sure that you are on a "level" at least equivalent to the contents up to, and including, note 8.

This note will be a very short note on Complex numbers.

fractions like 2/7, "floating" numbers with a "." (dot, or comma) like 4.56287, numbers like π (which never end behind the ".") etc.. etc..

Not entirely so, as we will see in a few minutes. But first, let's see what exactly we have right now, and try to classify them.

As you know, there exists a large

fractions like 1/3, 2/7 etc.. and "real" numbers which seems to include about "everything".

These classifications, are often viewed as infinite sets. The main sets are:

Here, we have positive numbers only.

They can be positive or negative.

Don'y forget. Even integers are "Rational numbers", since we can always represent them as a fraction, like 3/3, 10/2, 27/9 etc..

Note: there are a few more sets, but the most important ones are listed above.

These are all pretty "normal" numbers. There are many mathematical rules for operating on those numbers.

Let 'a' be one of such normal numbers. For example, we never have allowed that a

Really! In ordinary calculus or arithmetic, we have always assumed, or required, that a

The following holds (or is defined/required) for Imaginary numbers:

That is not compliant with the usual rules of engagement in arithmetic. Using numbers from

we always have required:

a x a ≥ 0 for any number we know in from

But, an "imaginary number" is

Thus we might have the number 3i, and the square of that number is (3i)

in effect, you may say that the square of bi is equal to -b

Pretty Strange isn't? We will understand it better when we see complex numbers.

Figure 1. Number sets

You should read figure 1 this way: Z encloses N (Z is bigger than N), but Q encloses Z, and R encloses everything.

Well everything? No, since the "imaginary numbers" are alien to the "normal" numbers, they are set apart.

In figure 1, you see the yellow ellips to illustrate that the are completely seperate from the rest.

Now, since scientists (like mathematicians and physicists) are completey "un-stoppable", they created a new set

of numbers: The complex numbers.

In figure 1, you see the square. In fact, it is

Now, if you take a real number (from R), say "a", and multiply it with an imaginary number, say "bi",

then we have nothing new, since a x bi = ci (where axb=c),

However, if you take a real number (from R), say "a" and

then we get a two dimensional construct "z":

to represent the real numbers along the x-axis (as we already always did), and represent the imaginary numbers

along the y-axis. So, along the x-axis we have the usual arrangement of real numbers, just as we have seen many times before

in all those plots of functions in former notes.

Along the y-axis, we have the "bi" numbers, where b is a real number (and can vary in the usual way),

and i is that strange object for which holds that i

The only point both axes intersect, is at (0,0) since then even an imaginary number is '0', if b=0.

Do we end up in something familiar? Yes, it all looks pretty much the same as the familiar XY plane we have seen so many times before.

So, instead of z=a+bi, we might just as well use "x" for "a" and "y" for "b", and so we have:

So we have complex numbers in the form of "z = x + iy". Note that both 'x' and 'y' are "real" numbers

The "iy" part, is the imaginary part. Indeed, it's the "i" which makes "iy" rather special.

Some obvious observations:

If y=0, then we have z = x + iy = x + i0 = x, which is just a common "real" number.

If x=0, then we have z = x + iy = 0 + iy = iy, which is just an imaginary number.

If x=0 ∧ y=0, then we have z=0. Viewed from the perspecive of the common XY plane,

z=0 corresponds to (0,0) and that corresponds to "the origin" (

Probably, the following is very important. In general, it is argued by almost all mathematicians,

that complex numbers are truly

than a special case of a complex number (namely, when y=0).

Complex numbers

Although complex numbers seem to make things more complex, it's really the other way around !

As an example: some horribly complex differential equations, or integrals, from the presepctive of "R",

are greatly simplyfied using

(ii):

In physics, since a long time, it's an essential element in most theories.

For example, Quantum Mechanics would not be the same theory without complex numbers.

Actually, there would not be such a theory at all.

My "focus" here is on "math", but in note 10, we will see some examples from physics.

For example, note the number z = 2+5i, and for example the number z =

You see? It's really easy to plot complex numbers in a XY coordinate system, if you know 'x' and 'y'.

Figure 2. Complex numbers represented in a XY coordinate system.

If you are familiar with 2 dimensional vectors, in a 2 dimensional Cartesesian coordinate system,

then indeed, it looks strikingly similar, but there are some differences.

If z=x+yi, you may, if you like, express it as point (x,y). However, keep in mind that "yi" is the

imaginary part of 'z', so that is not evident if you simply express 'z' as "(x,y)".

So, usually, it's not done that way, unless it's "handy" for some situation.

also note the number 2x-5i. That one is called the

This fact can be important in later notes. The complex conjugate is often written as z with a small "-"

above the letter, or with an asterix as a superscript, like z

In general:

If z=x+yi then it's complex comjugate is z

Note that if z=z

In fig. 2.2 you see a picture that illustrates, that "x" and "y" and "r", always form a right angled triangle.

Here, the 'x' and 'y' then form 2 sides of that right angled triangle, and the hypotenuse is 'r'.

The "length" of r, is the "absolute value" (or it's length, or the distance to "O") of the complex number.

We can apply the "Pythagorean theorem" here, which states:

If needed, please check note 4 again, because I think this is very important to understand.

Thus:

= r

and thus we again have cos

In general, people say that the lenght, or absolute value of the number z, notated as |z|, is:

x = rcos(α)

y = rsin(α)

Thus:

We have not dealt with anything else than "Cartesian coordinates" yet, which is like the XY plane having perpendicular

x- and y-axes. Indeed, you can perfectly "pinpoint" any number "z", by specifying the 'x' and 'y' values of z=x+yi.

There exists ofcourse a number of ways to mathematically "prove" Euler's equation. We are not going to do that yet,

since I will do it as a "special case" when we have a note on "coordinate transformations".

In that note (on "coordinate transformations"), the equation will get clear in a more natural fashion.

Here is Euler's equation. You only have take notice of the fact that it exists, and it's often used in many other derivations.

- z=x+yi (Cartesian coordinates: the real values x and y fully pinpoint the number "z").
- z= r e
^{iφ}(the distance "r" of "z" to the Origin, and the angle between "r" and the x-axis,

fully pinpoint the number "z".

- z = r cos(φ) + r sin(φ) i (which was shown in section 2.3).

Figure 3. Different ways to describe, or pinpoint, "z".

Ofcourse, there is much more to say on "complex numbers".

However, in my "series of notes", this will do for now.

The next note is a quick intro in "differential equations".